Question: The Southfork Feed company makes a feed mix from four ingredients-oats, corn, soybeans, and a vitamin supplement. The company has 300 pounds of oats, 400 of corn, 200 pounds of soybeans, and 100 pounds of vitamine supplement available for the mix. The company has the following requirements for the mix:

*At least 30% of the mix must be soybeans

*At least 20% of the mix must be the vitamin supplement.

*The ratio of corn to oats cannot exceed 2 to 1.

*The amount of oats cannot exceed the amount of soybeans.

*The mix must be at least 500 pounds.

A pound of oats costs &0.50; a pound of corn, $1.20; a pound of soybeans,$0.60; and a pound of vitamin supplement, $2.00. The feed company wants to know the number of pounds of each ingredient to put in the mix in order to minimize the cost.

a-formulate a linear programming model for this problem.

b-Solve the model by using the computer.

Constraints – The second sentence lists 4 resource constraints. The 5 bullets below that list the mixture constraints. The 4 resources constraints are simple, each of the decision variable values cannot exceed a specific limit. The first three of the five bullets are ratio constraints. You will need to convert these into linear inequalities. I will walk you through the first bullet. Let X1, X2, X3, and X4 be the number of pounds of oats, corn, soybeans, and vitamin supplement, respectively. If at least 30% of the mix must be soybeans, this means the ratio of soybeans to the entire mixture must be greater than or equal to 0.30 as written below.



X3 > 0.30 We convert this into a linear expression
X1 + X2 + X3 + X4 Multiply both sides by (X1 + X2 + X3 + X4)

X3 > 0.30(X1 + X2 + X3 + X4)

X3 > 0.30 X1 + 0.30X2 + 0.30X3 + 0.30X4 Subtract X3 from both sides

0 > 0.30 X1 + 0.30 X2 – 0.70 X3 + 0.30 X3

0.30 X1 + 0.30 X2 – 0.70 X3 + 0.30 X3 < 0 Re-written with parameter on RHS.

The last two bullets are linear inequalities.

You will therefore have 4 decision variables and 9 constraints, and you need to minimize cost.

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